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Note on beta elements in homotopy, and an application to the prime three case


Author: Katsumi Shimomura
Journal: Proc. Amer. Math. Soc. 138 (2010), 1495-1499
MSC (2010): Primary 55Q45
DOI: https://doi.org/10.1090/S0002-9939-09-10190-9
Published electronically: December 8, 2009
MathSciNet review: 2578544
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Abstract: Let $ S^0_{(p)}$ denote the sphere spectrum localized at an odd prime $ p$. Then we have the first beta element $ \beta_1\in\pi_{2p^2-2p-2}(S^0_{(p)})$, whose cofiber we denote by $ W$. We also consider a generalized Smith-Toda spectrum $ V_r$ characterized by $ BP_*(V_r)=BP_*/(p,v_1^r)$. In this note, we show that an element of $ \pi_*(V_r\wedge W)$ gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of $ \beta_{9t+3}$ at the prime three to complete a conjecture of Ravenel's: $ \beta_{s}\in \pi_{16s-6}(S^0_{(3)})$ exists if and only if $ s$ is not congruent to $ 4$, $ 7$ or $ 8$ mod $ 9$.


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Additional Information

Katsumi Shimomura
Affiliation: Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan
Email: katsumi@kochi-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-09-10190-9
Received by editor(s): April 19, 2009
Published electronically: December 8, 2009
Communicated by: Brooke Shipley
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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